Create a Muriel-like language and implementation

Question

Muriel is a language where the only way to loop is to create a quine and execute it. Your job is to create your own language with this property and an implementation of it in a pre-existing language.

We will define a Muriel-like language like as follows:

1. It has a built-in command for executing code in its own language, similar to Muriel's @ command or python's exec.
2. It is Turing complete.
3. If you remove the built-in execution command it is no longer Turing complete.

Muriel is Muriel-like with @ as its built-in execution command. Smurf is also Muriel-like (its built-in execution command is x). Python is not Muriel-like, since it is still Turing complete if you remove exec.

Your job is to create a Muriel-like programming language, and then to provide an implementation of it in a language of your choice. Your score is the length of the implementation, which you are trying to minimize.

Notes:

1. Your built-in execution does not need to function exactly like Muriel. It is up to you how it handles termination of the child program, errors, scoping, etc...
2. The language you write the implementation can not itself be Muriel-like. (This is so that @ is not a valid entry.) It also must have existed before this post.

Answer 1

Python 3 and "The minimum TC subset of Underload but with the characters changed to 2019, reversed, and with a 0 on top", 156 bytes

c=input()
n=1
t=c
s=[]
while c:
 *c,i=c
 if n:
  if"6">i:n+=int(i)-1
  if n:t=[i]+t
  else:s+=[t];t=[]
 elif"2"==i:n=1
 elif"2">i:s+=s[-1:]
 else:c+=s.pop()

Try it online!

Explanations in the header, commented code in the footer.